The challenge of Tikhonov regularization is selecting an optimal intermediate value of the regularization parameter \(\gamma \)  to retrieve the most accurate temperature distributions without knowing the true temperature distribution. Out of several \(\gamma \) -selection methods [1]}, [2]}, the L-curve method depicted in Figure REF  selects an accurate temperature \(CDF^{-1}\)  for all four test cases. This L-curve method works as follows. First the 2nd-order length-binning inversion is calculated for several values of \(\gamma \)  spanning several orders of magnitude, using a simulated annealing algorithm based on Corana et al [46] to determine the minimum of Eq. REF . Then the two terms of the minimization function, measurement residual and temperature gradient, are plotted for each \(\gamma \)  on the two axes of the L-curve in Figure REF b. The corresponding temperature distributions are shown as monotonically-increasing discrete solutions in Figure REF a. The small-\(\gamma \)  solutions appear in blue as chaotic and sharp-cornered in the temperature distribution panel, and as high-gradient, low-residual solutions on the L-curve panel. The large-\(\gamma \)  solutions appear in green as uniform and parabolic distributions on the temperature distribution panel, and as low-gradient, high-residual solutions on the L-curve panel. The best solution lies at the corner of the L-curve invtext3, hansentext, determined by a rectangular distance equation described in the appendix. This \(\gamma \) -selection is justified by Occam’s Razor, whereby of all the solutions with nearly equivalent fits to the linestrength data (those points which lie upon the vertical portion of Figure REF b), the simplest solution with the smallest temperature gradients is the most likely. Indeed, even though the true temperature \(CDF^{-1}\)  was not used in the L-curve inversion method, Figure REF  shows that the corner of the L-curve corresponds to the most accurate solution. In situations where the corner of the L-curve may be smoother and more ambiguous, one can select a range of plausible \(\gamma \)  and incorporate this spread in the temperature \(CDF^{-1}\)  into the uncertainty bars of the inversion.
